Schrodinger Bridges classical and quantum evolution

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1 Schrodinger Bridges classical and quantum evolution Tryphon Georgiou University of Minnesota! Joint work with Michele Pavon and Yongxin Chen! Newton Institute Cambridge, August 11, 2014

2 Positive contraction mappings for classical and quantum Schrodinger systems Tryphon T. Georgiou, Michele Pavon uploaded last night Optimal steering of a linear stochastic system to a final probability distribution Yongxin Chen, Tryphon T. Georgiou, Michele Pavon

3 History of Schrodinger bridges Bridges for Markov chains The Hilbert metric Bridges for quantum (TPTP) evolutions Bridges for Gauss-Markov process

4 Schrodinger 1931/32: The time reversal of the laws of nature! Kolmogoroff: The reversibility of the statistical laws of nature! Bernstein 1932 Fortet 1940 Beurling 1960 Jamison 1974/75 Follmer 1988! connections to Nelson s stochastic mechanics Zambrini, Wakolbinger, Dai Pra, Pavon,Ticozzi and others

5 Hilbert metric:! Hilbert 1895 Birkhoff 1957 Bushell 1973! Sepulchre, Sarlette, Rouchon 2010 Reeb, Kastoryano, Wolf 2011!

6 Schrodinger 1931/1932: suppose a large number of Brownian particles observed at two different times to evolve between two empirical distributions. What is the most likely intermediate distribution at any point in time?

7 Given initial and final distribution p 0 (x), p T (x) and transition p(x, y) Schrödinger hypothesised that p T ( ) 6= Z p(x, )p 0 (x)dx =: 0T (p 0 (x)) 0t : q 0 (x)! q = 1 2 2, q(0,x)=q 0 (x). Large deviations Sample paths Relative entropy Stochastic Control

8 Schrödinger system discretized time, space, N-particles Stirling s approximation optimized, lagrange multipliers the most likely joint density and transition probability P? (x 0,x T )= ˆ(x 0 )p(x 0,x T ) (x T ) and p? (x 0,x T )=p(x 0,x T ) (x T ) (x 0 ) Z p 0 (x 0 ) = ˆ(x 0 ) Z p T (x T ) = (x T ) p(x 0,x T ) (x T )dx T = ˆ(x 0 ) p(x 0,x T ) ˆ(x 0 )dx 0 = (x T ) 0(x 0 ) z } { 0,T (x T ) {z } T (x T ) ˆT (x T ) z } { 0,T ˆ(x0 ) {z } ˆ0(x 0 ) and p t (x t )= ˆt(x t ) (x t ) where t(x t ):= t,t (x T) ˆt(x t ):= 0,t ˆ(x0 )

9 Schrödinger system Schrödinger: there exists a solution except possibly for very nasty p 0, p T because the question leading to the pair of equations is so reasonable. Existence/uniqueness Fortet 1940 Résolution d un system d equations de M. Schr dinger Beurling 1960 An automorphism of product measures

10 Markov chains {1,...,N} states, x =(x 0,x 1,...,x T ) sample path t stochastic matrices, t 2 {1,...,T} P 2 probability induced by s on {1,...,N} T +1 P (x 0,...,x T )=P (x 0,x T )P (x 1,...,x T 1 x 0,x T ) Schrödinger question given p 0, p T p T 6= T 1 p 0 find Q(x 0,...,x T )=Q(x 0,x T )Q(x 1,...,x T 1 x 0,x T ) such that P P x T Q(x 0,x T )=p 0 (x 0 ) x 0 Q(x 0,x T )=p T (x T ) and minimizes the relative entropy X Q log Q P = X Q(x 0,x T ) log Q(x 0,x T ) P (x all x 0,x 0,x T ) +X T all Q( x 0,x T ) log Q( x 0,x T ) P ( x 0,x T ) Q(x 0,x T )

11 Lagrangian L(Q) = X Q(x 0,x T )log Q(x 0,x T ) P (x x 0,x 0,x T ) T X (x 0 X x 0 xt Q(x 0,x T ) p T (x T ) A + X x T µ(x T ) X x 0 Q(x 0,x T ) p 0 (x 0 )! (x 0 ) ˆ0 µ(x T ) T such that with = T 1

12 Schrödinger system if there is a solution ˆT = ˆ0 0 = T p 0 = 0 ˆ0 p T = T ˆT? = D T D 1 0 [Q(x 0,x T )] x0,x T = D T D ˆ0

13 Hilbert metric S real Banach space K closed solid cone in S x y, y x 2 K, M(x, y) := inf{ x y} define the Hilbert metric: m(x, y) := sup{ y x}. d H (x, y) :=log M(x, y) m(x, y) Examples: i) positive cone in R n ii) positive definite Hermitian matrices.

14 d H -gain bound of positive maps is a positive map: : K\{0}! K\{0}. Projective diameter ( ) :=sup{d H ( (x), (y)) x, y 2 K\{0}} Contraction ratio, orgain/h-norm k k H := inf{ d H ( (x), (y)) apple d H (x, y), for all x, y 2 K\{0}}.

15 Birkho -Bushell theorem Let positive, monotone, homogeneous of degree m, i.e., x y ) (x) (y), and then ( x) = m (x), k k H apple m. For the special case where is also linear, the (possibly stronger) bound k k H =tanh( 1 4 ( )) also holds.

16 Solution of the Schrödinger system Lemma Let > e 0(element-wisepositive)stochasticmatrix p 0, p T probability vectors then k k H < 1. proof i) ( ) =sup{d H ( (x), (y)) x, y 2 K\{0}} remains the same if we restrict x, y to be probability vectors ii) d H ( (x), (y)) < 18x, y. iii) the probability simplex is compact.

17 Solution of the Schrödinger system Consider ˆ' 0! ˆ'T ˆ' 0 (x 0 )= p 0(x 0 ) ' 0 (x 0 ) " # ' T (x T )= p T (x T ) ˆ' T (x T ) where ' 0 ' T D T : ' 0 7! ˆ' 0 (x 0 )= p 0(x 0 ) ' 0 (x 0 ) D T : ˆ' T 7! ' T (x T )= p T(x N ) ˆ' T (x T ) are componentwise division of vectors ) d H -isometries! The composition ˆ' 0! ˆ'T D T! ' T! ' 0 D 0! ( ˆ' 0 ) next is strictly contractive in the Hilbert metric.

18 Sinkhorn s theorem If > e 0, then 9 a i, b j such that [ ij a i b j ] i,j doubly stochastic. Cf. p 0 = 1, p T = 1? = D T D 1 doubly stochastic 0 i.e., (? ) 1 = 1 but also (? )1 = 1

19 Quantum analogues Density matrices: D = { 0 trace( ) =1} TPTP: E : D! D :! = P n E i=1 E i E i with i.e., E (I) =I n E X i=1 E i E i = I E is positivity improving: if 0 ) E( ) > 0

20 Reference quantum evolution TPCP maps {E t ;0apple t apple T 1} with Kraus representation E t : t 7! t+1 = X i E t,i t E t,i, t =0, 1,...,T 1. Consider the composition E 0:T := E T 1 E 0. initial and a final 0 and T Problem Find F 0:T = F T 1 F 0 such that and F close to E F 0:T ( 0 )= T.

21 rank-1 corrections F t ( ) = t+1 E t ( 1 t ( ) t ) t+1 1 i.e., F t = t+1 E t t where are rank-1 Kraus maps, n =1 Corresponds to the commutative case via: =

22 Quantum version of Sinkhorn s thm Suppose E 0:T is positivity improving Then, 9 observables 0, T such that, for any factorization the map 0 = 0 0, and T = T T, F( ) := T E 0:T ( 1 0 ( ) 0 ) T is a doubly stochastic Kraus map, in that F(I) =I as well as F (I) =I.

23 Proof ˆ0 E 0,T! ˆT ˆ0 = 1 0 " # T = ˆ 1 T The composition map C : ˆ0 starting 0 E 0,T! ˆT E 0,T T ( ) 1 E 0,T ( ) 1! T! 0! ˆ0 next is strictly contractive the steps are identical

24 General case Given E 0:T and 0 and T if 9 0, T, ˆ0, ˆT solving E 0:T ( T) = 0, E 0:T ( ˆ0) = ˆT, Then, for any factorization the map 0 = 0 ˆ0 0, T = T ˆT T. 0 = 0 0, and T = T T, F( ) := T E 0:T ( 1 0 ( ) 0 ) T is a quantum bridge for (E 0:T, 0, T ), namely F(I) =I and F ( 0 )= T.

25 Conjecture The quantum Schrödinger system has a solution for arbitrary 0, T Snag in the proof:! ˆ and ˆ! are not isometries, e.g., 1/2 2 D T : ˆT 7! T = 1/2 T 1/2 ˆ 1 T 1/2 1/2 T T ˆD 0 : 0 7! ˆ0 =( 0 ) 1/2 ( 0 ) 1/2 Extensive numerical evidence that the composition has a fixed point Software for numerical experimentation bridge/

26 Pinned bridge E 0:T positivity improving and two pure states 0 = v 0 v 0 and T = v T v T (i.e., v 0,v T are unit norm vectors), define and F ( ) := 1/2 T E ( 0 := E(v T v T ) T := v T v T, 1/2 0 ( ) 1/2 0 ) 1/2 T 1/2 (where, clearly, T = T = v T v T ). Then, F is TPTP and satisfies the marginal conditions T = F ( 0 ).

27 Example E 1 = E( ) =E 1 ( )E 1 + E 2( )E 2 + E 3( )E 3 # " # 2 q ,E 2 = q,e 1 3 = 4q " # " # 1/4 0 2/3 0 0 = and 1 = 0 3/4 0 1/3 "q = 1 = ˆ0 = ˆ1 = " # 1/2 0 " 0 1/2 # 2/3 0 " 0 1/3 # 1/2 0 " 0 3/2 # F 1 = " p 2/ #, F 2 = " # " p # /3 p, F 3 p 0 1/3 1/3 0

28 Recap Hilbert metric ) constructive existence proofs for i) classical Schrödinger systems ii) quantum Sinkhorn version (uniform marginals) iii) general case open Final topic: Schrödinger bridges for degenerate classical linear stochastic systems anewtypeofoptimalcontrolproblem

29 Optimal steering of state-densities min relative entropy $ Girsanov $ minimum energy stochastic control dx = bdt + dw di usion dx =(b + u)dt + dw controlled di usion min{e{kuk 2 } p 0,p T } relative entropy from prior (dai Pra) our interest: inertial particles, cooling of oscillators dx = vdt dv =(b + u)dt + dw controlled degenerate di usion

30 Optimal steering of state-densities dx(t) = A(t)x(t)dt + B(t)u(t)dt + B(t)dw(t) Given initial and terminal (target) Gaussian densities with covariances 0, T. Find u(t) witht 2 [0,T]thatsteersthesystem from the initial to the target state density and minimizes Z T E{ 0 u(t) 0 u(t)dt}

31 Optimal steering of state-densities Theorem (Gauss-Markov Schrödinger bridge): There exists a unique solution to the following (analogue of the Schrödinger system) Q(T ), P (0) values for matrices satisfying 1 0 = Q(0) 1 + P (0) 1 1 T = Q(T ) 1 + P (T ) 1 and Q(0), P (T )obtainedvia Q(t) =A(t)Q(t)+Q(t)A(t) 0 + B(t)B(t) 0 P (t) =A(t)P (t)+p (t)a(t) 0 B(t)B(t) 0 with Q(t) invertibleover[0,t].

32 The optimal control is u(t) = B(t) 0 Q(t) 1 x(t) The controlled degenerate di usion is the closest to the uncontrolled di usion in the relative entropy sense. Q(0) = N(T,0) 1/2 S 1/2 0 S I S 1/2 S 0 T S 1/ I 1/2! 1 S 1/2 0 N(T,0) 1/2 N(T,0) is the controllability Grammian.

33 Gauss Markov model for inertial particles dx(t) = v(t)dt dv(t) = u(t)dt + dw(t) ntrol force at our disposal, re

34 Gauss Markov model for inertial particles

35 Gauss Markov model for Nyquist-Johnson noise driven oscillator Ldi L (t) = v C (t)dt RCdv C (t) = v C (t)dt Ri L (t)dt + u(t)dt + dw(t) th all parameters in compatible units. Withou

36 Thank you for your attention